Glassy systems are characterized by an extremely sluggish dynamics without
any simple sign of long range order. It is a debated question whether a correct
description of such phenomenon requires the emergence of a large correlation
length. We prove rigorous bounds between length and time scales implying the
growth of a properly defined length when the relaxation time increases. Our
results are valid in a rather general setting, which covers finite-dimensional
and mean field systems.
As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin
glass models on random regular graphs. We present the first proof that a model
of this type undergoes a purely dynamical phase transition not accompanied by
any thermodynamic singularity.Comment: 24 pages, 3 figures; published versio